## Journal of Symbolic Logic

### Les Automorphismes D'un Ensemble Fortement Minimal

Daniel Lascar

#### Abstract

Let $\mathfrak{M}$ be a countable saturated structure, and assume that $D(\nu)$ is a strongly minimal formula (without parameter) such that $\mathfrak{M}$ is the algebraic closure of $D(\mathfrak{M})$. We will prove the two following theorems: Theorem 1. If $G$ is a subgroup of $\operatorname{Aut}(\mathfrak{M})$ of countable index, there exists a finite set $A$ in $\mathfrak{M}$ such that every $A$-strong automorphism is in $G$. Theorem 2. Assume that $G$ is a normal subgroup of $\operatorname{Aut}(\mathfrak{M})$ containing an element $g$ such that for all $n$ there exists $X \subseteq D(\mathfrak{M})$ such that $\operatorname{Dim}(g(X)/X) > n$. Then every strong automorphism is in $G$.

#### Article information

Source
J. Symbolic Logic, Volume 57, Issue 1 (1992), 238-251.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743903

Mathematical Reviews number (MathSciNet)
MR1150937

Zentralblatt MATH identifier
0785.03018

JSTOR